We calculate the E-polynomials of certain twisted GL(n, C)-character varieties M n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n, F q ) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n, C)-character variety. The calculation also leads to several conjectures about the cohomology of M n : an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.

We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GL n (C) with fixed generic semi-simple conjugacy classes at k punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of GL n (F q ). We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincaré polynomial of certain associated quiver varieties which we obtain using the character table of gl n (F q ). In the last main result we prove that the Poincaré polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of GL n (F q ). As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, typically wild, quivers and the representation theory of GL n (F q ).

We study ζ-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The ζ-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to 'see' these curves in the geometry of the quintic. Having these ζ-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the ζ-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the ζ-functions are rational functions and the degrees of the numerators and denominators are exchanged between the ζ-functions for the manifold and its mirror. It is clear nevertheless that the ζ-function, as classically defined, makes an essential distinction between Kähler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a 'quantum modification' of the ζ-function that restores the symmetry between the Kähler and complex structure parameters. We note that the ζ-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.

Abstract. An explicit formula is derived for the logarithmic Mahler measure m(P) of P(x, y) = p(x)y− q(x), where p(x) and q(x) are cyclotomic. This is used to find many examples of such polynomials for which m(P) is rationally related to the Dedekind zeta value ζ F (2) for certain quadratic and quartic fields.

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomasinvariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich-Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isoytpical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincaré polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties. The main resultsLet Γ = (I, Ω) be a quiver: that is, an oriented graph on a finite set I = {1, . . . , r} with Ω a finite multiset of oriented edges. In his study of the representation theory of quivers, Kac [17] introduced A v (q), the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field F q of dimension v = (v 1 , . . . , v r ) and showed they are polynomials in q. We call A v (q) the Kac polynomial for Γ and v. Following ideas of Kac [17], Hua [16] proved the following generating function identity:where P denotes the set of partitions of all positive integers, Log is the plethystic logarithm (see [14, §2.3.3]), , is the pairing on partitions defined bywith m j (λ) the multiplicity of the part j in the partition λ, which implies positivity for certain refined DT-invariants of symmetric quivers with no potential.The goal of Kontsevich-Soibelman's theory is to attach refined (or motivic, or quantum) DonaldsonThomas invariants (or DT-invariants for short) to Calabi-Yau 3-folds X. The invariants should only depend on the derived category of coherent sheaves on X and some extra data; this raises the possibility of defining DTinvariants for certain Calabi-Yau 3-categories which share the formal properties of the geometric situation, but are algebraically easier to study. The simplest of such examples are the Calabi-Yau 3-categories attached to quivers (symmetric or not) with no potential (c.f. [12]).Denote by Γ = (I, Ω) the double quiver, that is Ω = Ω Ω opp , where Ω opp is obtained by reversing all edges in Ω. The refined DT-invariants of Γ (a slight renormalization of those introduced by Kontsevich and Soibelman [19]) are defined by the following combinatorial construction.In fact, as a consequence of Efimov's proof [10] of [20, Conjecture 1], DT v (q) actually has non-negative coefficients. We will give an alternative proof of this in (1.10) by interpreting its coefficients as dimensions of cohomology groups of an associated quiver variety. Remark 1.2. We should stress that we have restricted to double quivers for the benefit of exposition; our results extend easily to any symmetric quiver. We outline how to treat the general case in §3.2. The technical starting point in this paper is a common generalization of (1.3) and Hua'...